Theorem 19.33 1413

Description: Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
19.33	⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓))

Proof of Theorem 19.33

Step	Hyp	Ref	Expression
1		orc 665	. . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
2	1	alimi 1384	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜑 ∨ 𝜓))
3		olc 664	. . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
4	3	alimi 1384	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 ∨ 𝜓))
5	2, 4	jaoi 668	1 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓))

Syntax hints:   → wi 4   ∨ wo 661  ∀wal 1282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-gen 1378

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  19.33b2  1560